If the tag question for ¨I am free¨ is ¨aren´t I?¨. Which is the tag question for ¨I am not at home¨?

Answer 1

The tag is there to confirm your idea "I am free". so you are not asking yourself, you are asking another person, so the tag evolved as "aren't you, too, free".
Then, the phonetics of English do not support the "ain't I" or "amt I" so the "aren't" developed because it sounds better. (Many, many word constructions evolved simply because they sounded better than something else.) Your second one follows the regular rules. the tag is "...am I?" because your idea is negative (I am not at home.) so the tag has to be the opposite, that is, positive.
Also, the "I am not at home?" is more of a philosophical question that may or may not involve a tag. It is very, very close to the famous line in Wizard of Oz "We aren't in Kansas anymore." This was NOT a question, and therefore did not need a tag. It was a statement about Dorothy's reality. If she had added the tag "...are we" the line would not be famous today.

Answer 2

Counter question: As the pronoun "I" demands the verb correspondent "Am" would not the appropriate tag to
"I am free?" be "Am I not?" which would therefore be the same for the second?

Answer 3

Isn't it still "Aren't I?" Because it's a contraction saying "Are not I?" or "Am I not?" It still should be "Aren't I" ....I don't know what I'm saying, lol

Answer 4

Where I be?

Answer 5

am i not

Answer 6

am I?

Answer 7

am I?

Answer 8

am I?

Answer 9

",am I?"

Answer 10

am I...